The Sobolev critical scaling regularity for the quadratic derivative nonlinear wave equation in 2D is $s_c=1$, however the best known Sobolev space well-posedness result is for Cauchy data in $H^s$ with $s>7/3$. Following Grunrock's 3D result for the quadratic derivative NLW, we consider initial data in the Fourier-Lebesgue spaces $hat{H}_s^r$, which coincide with the Sobolev spaces of the same regularity for $r=2$, but scale like lower regularity Sobolev spaces for $15/4$ is known to be ill-posed. With an additional trilinear estimate, this also yields almost-critical well-posedness for the Ward map problem.